Question

Prove the identity. \[ \sin ^{2} x\left(1+\cot ^{2} x\right)=1 \] Note that each Statement must be based on a Rule chosen from the Rule menu. To vee a detalled description of a Rule, select the Moin I"

Answer 1

To prove the trigonometric identity sin^2x(1+cot^2x)=1, one can follow the Pythagorean identity sin^2x + cos^2x = 1 and the definition of cotangent in terms of sine and cosine, simplifying the expression to the Pythagorean identity.

Step-by-step explanation:

To prove the identity sin2x(1+cot2x)=1, we can start by using the Pythagorean identity which states that sin2x + cos2x = 1. Next, we know that cotangent is the reciprocal of tangent, which is sin over cos, so cot2x = cos2x/sin2x. When you multiply sin2x with cot2x, the sin2x terms will cancel out and you'll be left with cos2x.

Putting this into the original expression: sin2x + sin2x(cot2x) = sin2x + cos2x = 1

Thus, sin2x(1+cot2x) simplifies to the Pythagorean identity, which proves it equals 1.